Valuation Of Financial Derivatives Research Articles

Deep Equal Risk Pricing of Financial Derivatives with Non-Translation Invariant Risk Measures

The objective is to study the use of non-translation invariant risk measures within the equal risk pricing (ERP) methodology for the valuation of financial derivatives. The ability to move beyond the class of convex risk measures considered in several prior studies provides more flexibility within the pricing scheme. In particular, suitable choices for the risk measure embedded in the ERP framework, such as the semi-mean-square-error (SMSE), are shown herein to alleviate the price inflation phenomenon observed under the tail value at risk-based ERP as documented in previous work. The numerical implementation of non-translation invariant ERP is performed through deep reinforcement learning, where a slight modification is applied to the conventional deep hedging training algorithm so as to enable obtaining a price through a single training run for the two neural networks associated with the respective long and short hedging strategies. The accuracy of the neural network training procedure is shown in simulation experiments not to be materially impacted by such modification of the training algorithm.

Reflective Study of a Biographical Journey and Critical Analysis of Contributions to Problematic Valuation of Financial Derivatives (2Nd PhD Manuscript)

Reflective Study of a Biographical Journey and Critical Analysis of Contributions to Problematic Valuation of Financial Derivatives (2Nd PhD Manuscript)

Approximation of single‐barrier options partial differential equations using feed‐forward neural network

AbstractArtificial neural networks are generally employed in the numerical solution of differential equation problems. In this article, we propose an approach that deals with the combination of the feed‐forward neural network method and the optimization technique in solving the partial differential equation arising from the valuation of barrier options. The methodology entails transforming the extended Black–Scholes partial differential equations (PDE), which defines a barrier option, into a constrained optimization problem, and then proposing a trial solution that reduces the differential equation problem to an unconstrained one. This trial function consists of the adjustable and non‐adjustable neural network parameters. We design it to be differentiable, analytic, and satisfy the initial and boundary conditions of the corresponding option pricing PDE. We compare the corresponding option values to the Monte‐Carlo simulated values, Crank–Nicolson finite‐difference values and the exact Black–Scholes prices. Numerical results presented in this research show that neural networks can sufficiently solve PDE‐related problems with sufficient precision and accuracy. Furthermore, they can be applied in the fast and accurate valuation of financial derivatives without closed analytic forms.

Direct Solution of the Black-Scholes PDE Models with Non-Integer Order

This paper presents a direct solution of Black-Scholes PDE models with non-integer order via the non-integer iterative method. The Black-Scholes PDE models of non-integer order became popular and accepted globally by option traders for the valuation of financial derivatives over the period of time and the study of its numerical approaches has a wide range of applications in the real world.. The performance of the method has been confirmed and measured.

Penalty Methods for Bilateral XVA Pricing in European and American Contingent Claims by a PDE Model

Taking into account default risk in the valuation of financial derivatives has become increasingly important, especially after the 2007-2008 financial crisis. Under some assumptions, the valuation of financial derivatives, including a value adjustment to account for default risk (the so-called XVA), gives rise to a nonlinear PDE. We propose numerical methods for handling the nonlinearity in this PDE, the most efficient of which are discrete penalty iteration methods. We first formulate a penalty iteration method for the case of European contingent claims, and study its convergence. We then extend the method to the case of American contingent claims, resulting in a double-penalty iteration. We also propose boundary conditions and their discretization for the XVA PDE problem in the cases of call and put options, as well as the case of a forward contract. Numerical results demonstrate the effectiveness of our methods.

Penalty methods for bilateral XVA pricing in European and American contingent claims by a partial differential equation model

Accounting for default risk in the valuation of financial derivatives has become increasingly important, especially since the 2007–8 financial crisis. Under some assumptions, the valuation of financial derivatives, including a value adjustment to account for default risk (the so-called XVA), gives rise to a nonlinear partial differential equation (PDE). We propose numerical methods for handling the nonlinearity in this PDE, the most efficient of which are the discrete penalty iteration methods. We first formulate a penalty iteration method for the case of European contingent claims and study its convergence. We then extend the method to the case of American contingent claims, which results in a double-penalty iteration. We also propose boundary conditions and their discretization for the XVA PDE problem in the case of a call option, a put option and a forward contract. Numerical results demonstrate the effectiveness of our methods.

The Valuation of Financial Derivatives Subject to Counterparty Risk and Credit Value Adjustment

The Valuation of Financial Derivatives Subject to Counterparty Risk and Credit Value Adjustment

Bilateral Defaultable Financial Derivatives Pricing and Credit Valuation Adjustment

The one-side defaultable financial derivatives valuation problems have been studied extensively, but the valuation of bilateral derivatives with asymmetric credit qualities is still lacking convincing mechanism. This paper presents an analytical model for valuing derivatives subject to default by both counterparties. The default-free interest rates are modeled by the Market Models, while the default time is modeled by the reduced-form model as the first jump of a time-inhomogeneous Poisson process. All quantities modeled are market-observable. The closed-form solution gives us a better understanding of the impact of the credit asymmetry on swap value, credit value adjustment, swap rate and swap spread.

Importance sampling for jump processes and applications to finance

Adaptive importance sampling techniques are widely known for the Gaussian setting of Brownian driven diffusions. In this work, we want to extend them to jump processes. Our approach relies on a change of the jump intensity combined with the standard exponential tilting for the Brownian motion. The free parameters of our framework are optimized using sample average approximation techniques. We illustrate the efficiency of our method on the valuation of financial derivatives in several jump models.

Another Look at Option Valuation Based on Arithmetic Brownian Motion

We take another look at arithmetic Brownian motion as a framework for developing financial derivatives valuation and risk management tools. We compare and contrast option valuation based on geometric Brownian motion and arithmetic Brownian motion. An alternative way to handle negative stock prices within the arithmetic Brownian motion approach is identified that is more consistent with empirical observation than the traditional approach of assuming no possibility of zero prices within the geometric Brownian motion. After comparing a wide array of sensitivities with both models, we discuss numerous strengths and weakness of both approaches. Option valuation based on arithmetic Brownian motion appears to be a credible additional tool for financial quantitative analysis.

Valuing Securities and Managing Portfolios Under Uncertainty: A Reminder of the Underlying Assumptions

Valuation of financial derivatives is essential in the measurement and management of risk hence should be well understood. The bedrock for this appreciation, however, is the concept of costless arbitrage and frictionless markets. This paper also discusses discount and coupon bearing bonds in terms of their similarities and differences. Following on this, the importance of the Dividend Growth model, and its failure to win universal acceptance in stock valuation is mentioned. Above all, the paper notes the fact that investors are risk averse, but willing to bear more risk for more return. Noteworthy, for any level of risk investors want more returns; while for any given level of return they want less risk. In addition, the importance of the Capital Asset Pricing Model in formalising the relationship between a security and its level of risk is discussed. Finally, the paper reflects on the use of total risk based and systematic risk based metrics in measuring the performance of investment portfolios.

A framework for adaptive Monte Carlo procedures

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented in [Arouna, Monte Carlo Methods Appl. 10: 1–24, 2004, Arouna, The Journal of Computational Finance 7: Winter 2003/2004, Su and Fu, Journal of Computational Finance 5: 27–50, 2002, Vázquez-Abad and Dufresne, Accelerated simulation for pricing asian options: 1493–1500, IEEE Computer Society Press, 1998]. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives.

Parametric Properties of Semi-Nonparametric Distributions, with Applications to Option Valuation

We derive the statistical properties of the semi-nonparametric (SNP) densities of Gallant and Nychka (1987). We show that these densities, which are always positive, are more flexible than truncated Gram–Charlier expansions with positivity restrictions. We use the SNP densities for financial derivatives valuation. We relate real and risk-neutral measures, obtain closed-form prices for European options, and analyze the semiparametric properties of our pricing model. In an empirical application to S&P500 index options, we compare our model to the standard and Practitioner's Black–Scholes formulas, truncated expansions, and the Generalized Beta and Variance Gamma models.

Financial Derivatives Valuation

The article contains a thorough analysis of the world financial derivatives market with special reference to theoretical issues. It puts forward and traces Black-Scholes options valuation model. The author underlines particularities of advanced practical derivatives price calculation, using the math formula of the Nobel Prize winners. Some practical features of the concept and its implementation within notorious Long-Term Capital Management hedge fund are emphasized.

A continuous dependence result for ultraparabolic equations in option pricing

We prove continuous dependence results for solution to the Cauchy problem related to degenerate parabolic equations arising in the valuation of financial derivatives. These results are crucial in some standard calibration procedure for recent stochastic volatility and interest rates models.

Lie theory to value financial derivatives with time dependent parameters

We present a Lie algebraic technique, based upon the Wei-Norman Theorem, for the valuation of financial derivatives with time dependent parameters. Then we investigate upon the possibility to use results of Information Geometry in this Theorem to reduce the system of the pricing partial differential equations to a system of algebraic equations.

NUMERICAL SOLUTION OF TWO-FACTOR MODELS FOR VALUATION OF FINANCIAL DERIVATIVES

In this paper we propose a general methodology to solve partial differential inequalities arising in the valuation of financial derivatives. Firstly we show how to build up a weak formulation and how to discretize it by implicit finite difference schemes in time and finite elements in space and we propose an iterative algorithm to solve the discrete variational inequality. Although the methodology is quite general, here it is applied to solve two important two-factor models: one for valuation of dividend paying Amerasian options and another for convertible bonds with stochastic interest rate and call and put features. In both cases we validate our results with some others existing in the literature.

The Brownian Bridge Does Not Offer a Consistent Advantage in Quasi-Monte Carlo Integration

The Brownian bridge has been suggested as an effective method for reducing the quasi-Monte Carlo error for problems in finance. We give an example of a digital option where the Brownian bridge performs worse than the standard discretization. Hence, the Brownian bridge does not offer a consistent advantage in quasi-Monte Carlo integration. We consider integrals of functions of d variables with Gaussian weights such as the ones encountered in the valuation of financial derivatives and in risk management. Under weak assumptions on the class of functions, we study quasi-Monte Carlo methods that are based on different covariance matrix decompositions. We show that different covariance matrix decompositions lead to the same worst case quasi-Monte Carlo error and are, therefore, equivalent.

A TRANSACTION COST CONVERGENCE RESULT FOR GENERAL HEDGING STRATEGIES

This paper focuses on the valuation of financial derivatives with transaction costs. These financial products derive their value from other financial products, which are usually modeled as diffusion processes and called “underlyings”. The prices of such financial derivatives are generally written as the expectation of a function of the underlying process and can therefore be written as solutions of Partial Differential Equations. If there are transaction costs in the market, we prove that the price of a derivative converges towards the solution of a non-linear Partial Differential Equation as these transaction costs go to zero and the frequency of their payment goes to infinity. Our result generalizes those of Leland (1985) and Henrotte (1994). It holds for derivatives that are functionals of the underlying process rather than just functions (so-called “path-dependent”). It also holds if other derivatives are used in lieu of underlyings, and for derivatives whose value is not supposed to be a convex function of the underlying's price.

Valuation of financial derivatives with time-dependent parameters: Lie-algebraic approach

Based upon the Wei-Norman theorem, this paper presents a Lie-algebraic technique for the pricing of financial derivatives with time-dependent parameters. By exploiting the dynamical symmetry of the pricing partial differential equations of the financial derivatives, the new method enables us to derive analytical closed-form pricing formulae very straightforwardly. We believe that this new approach will provide an efficient method for the valuation of financial derivatives.